Klimeck – ECE606 Fall 2012 – notes adopted from Alam
ECE606: Solid State Devices
Lecture 7Gerhard Klimeck
1
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Law of mass-action & intrinsic concentration •Statistics of donors and acceptor levels• Intrinsic carrier concentration•Temperature dependence of carrier concentration•Multiple doping, co-doping, and heavy-doping•Conclusions
2
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Carrier Distribution
( )cg E
Eυ
FE( )g Eυ
cE
( )1 f E−
( )f E
( ) ( )cg E f E
( ) ( )1g E f Eυ −
( ) ( )top
c
E
cEn g E f E dE= ∫
( ) ( )1bot
E
Ep g E f E dE
υ
υ = − ∫
DOS F-D concentrationE E
1
3
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Electron Concentration in 3D solids
( ) ( )
( )( )2 3
2 12
2 1 βπ −
=
−= ×
+
∫
∫ℏ
top
c
top
Fc
E
cE
* *E n n C
E EE
n g E f E dE
m m E EdE
e
( )3 2
2 1 2
0 1
22
*n
C
dF
e
mN
h ξ ηπ β ξ ξη
∞
−=
≡ +∫
( )( ) ( )2 3
2 1
1 β βπ − −
∞
+
−∫≃
ℏ c Fc c
*C
E
n
E EE E
*nm m
dEe e
E E
( ) ( )1 2
2 η η βπ
= ≡ −c c FC CF EN E
Assume wide bands
Include spinfactor of 2
4
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Boltzmann vs. Fermi-Dirac Statistics
( ) ( )1 2
23ηη η β
π= → − ≡ − >c
C c C c C Fn N F N e if E E
( ) ( )cg E f E
( ) ( )1g E f Eυ − FE
ηce ( )1 2 ηcF
5
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Effective Density of States
( ) ( )1 2
23c FE E
C c C c Fn N F N e if E Eβη βπ
− −= → − >
( ) ( )cg E f E
( ) ( )1g E f Eυ −
CN
VN
FEFE
As if all states are at a single level EC
6
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Law of Mass-Action
( )
( )
β
β
− −
+ −
=
=
c F
v F
E EC
E EV
n N e
p N eFE
( )β
β
− −
−
× =
=
c v
g
E EC V
E
C V
n p N N e
N N e
Product is independent of the Fermi level!Very useful balance equation! Will use it often
7
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Fermi-Level for Intrinsic Semiconductors
2β−=≡
gE
i
F
V
i
Cn e
E
N
E
NFE
( ) ( )
1
2 2
β β
β
− − + −= ⇒ =
= +
c i v iE E E E
Gi
V
C
VCn p e e
EE ln
N
N
N
N
E
k
3
2
2 β−
= =
= g
V
i
E
i CN
p n
n eN
n
8
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
• We discussed how electrons are distributed in electronic states
defined by the solution of Schrodinger equation.
• Since electrons are distributed according to their energy,
irrespective of their momentum states, the previously
developed concepts of constant energy surfaces, density of
states etc. turn out to be very useful.
=> will not discuss Schroedinger Eq. anymore
=> everything is captured in bandedges and effective masses
• We still do not know where EF is for general semiconductors … If
we did, we could calculate electron concentration.
Summary – DOS and Fermi Functions
9
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Law of mass-action & intrinsic concentration •Statistics of donors and acceptor levels• Intrinsic carrier concentration•Temperature dependence of carrier concentration•Multiple doping, co-doping, and heavy-doping•Conclusions
10
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
E-k diagram vs. band-diagram
c refP.E. E E qV= − = −
Eυ
cEKinetic energy
PotentialEnergy
11
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Position Resolved E-k Diagram
Eυ
cE
Eυ
cE
Eυ
cE
12
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
E-k Diagram vs. Band-diagram
( )c refP.E. E E qV x= − = −
refE
V=0 V=V1
13
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Potential, Field and Charge
c refqV E E− = −
refE
2
2
d d V
dx dxρ = =E
c refP.E. E E= −
1 cdV dE
dx q dx= − =E
In most practical cases start from charge and derive potentials!=> Useful to learn “graphical” integration
14
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Law of mass-action & intrinsic concentration •Statistics of donors and acceptor levels• Intrinsic carrier concentration•Temperature dependence of carrier concentration•Multiple doping, co-doping, and heavy-doping•Conclusions
15
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Carrier Distribution
( )cg E
Eυ
FE
( )g Eυ
cE
( )1 f E−
( )f E
( ) ( )cg E f E
( ) ( )1g E f Eυ −
( ) ( )top
c
E
cEn g E f E dE= ∫
( ) ( )1bot
E
Ep g E f E dE
υ
υ = − ∫
DOS F-D concentrationE E
1
Critical items here;Intrinsic semiconductor has VERY few active electronsni is of the order of 1010/cm3. In 1022/cm3 atoms!Do not include the coulomb interactions of individual free electronsOften good enough to forget about the Distribution of carriers in energy => Replace by delta functions at band edge
CN
VN
FE
16
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Effective Density of States
( ) ( ) ( )1 2
23βη β
π− −= → − >c FE E
C c C c Fn N F N e if E E
( ) ( )cg E f E
( ) ( )1g E f Eυ −
CN
VN
FEFE
As if all states are at a single level EC
Often good enough to forget about the Distribution of carriers in energy => Replace by delta functions at band edge 17
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
E-k Diagram vs. Band-diagram
CE
VE
V=0 V=V1
All quantum mechanics is now hidden in a single point per band!18
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Potential, Field and Charge
c refP.E. E E= −
c refqV E E− = −
refE
2
2
d d V
dx dxρ = =E
1 cdV dE
dx q dx= − =E
In most practical cases start from charge and derive potentials!=> Useful to learn “graphical” integration
19
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Law of mass-action & intrinsic concentration •Statistics of donors and acceptor levels• Intrinsic carrier concentration•Temperature dependence of carrier concentration•Multiple doping, co-doping, and heavy-doping•Conclusions
20
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Insulator, Semiconductor, Metal
• Metal:»Conducts electrons even at very low temperatures
»Fermi Level crosses multiple bands even at very low temperaturs
• Semiconductor»Very weakly conducting
»Si: Eg=1.1eV ni~1010/cm3 in 1023/cm3 0.1 in a trillion
»GaAs: Eg=1.42eV ni~106/cm3 in 1023/cm3
»Ge: Eg=0.8eV ni~1013/cm3 in 1023/cm3 0.1 in a billion
• Insulator»“Not” conducting
»SiO2, Eg= 9eV, ni~10-68/cm3 � The whole earch has about 1050 atoms! If you made the whole worl out of
glass there would be not one electon conductive at room temperature!2β−=
≡
gE
i
F
V
i
Cn e
E
N
E
N
21
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Simplified Planar View of Atoms
22
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Donor Atoms
Even with donors, material is charge neutral
23
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Donor Atoms in H2-analogy
= +r0
= r0
24
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Donor Atoms in Real and Energy Space
( ) 20
20
40
2
02 4
13
1
16
πε= −
= − ×
ℏ
*host
s,hos
*host
s,host
tm
m
m K
q m
K
m
.
r0
ET=E1
( )4
1 2
02 4πε= −
ℏ
*host
s ,host
q
KE
m
~10s meV
1/β~kBT~25meV at T=300K25
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Assumption of Large Radius …
20
1 2
4πε=
ℏs ,host,P *
host
Kr
m qr0
1
12 90 53 12 9
0 53= × =,P
.r . A . A
.
02
02
10
0
4πε=
=
ℏ s,host*host
s ,hostH *
host,
m K
m
K
m
q
r/ m
m
a~0.5nm=5A => hundreds of Si atoms
26
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Characteristics of Donor Atoms
The number of donor atoms is much smaller compared to host atoms. Therefore, the electrons from one donor atom can go to the other donor atoms only via the conduction /valence bands of the host crystal.
Just like a Hydrogen atom, it is possible to have multiple localized level for a given atom (see the blue levels).
Good donors live close to the conduction band, so that they can offer electrons easily. However, if they are below the midgap, the donor levels are marked with (D) to differentiate them from acceptor atoms (which live close to the valence band).
(D)
27
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Acceptor Atoms
Even with acceptor, material is charge neutral
28
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Characteristics of Acceptor Atoms
A
r0
ET
29
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Amphoteric Dopants
acceptor-typeDonor-type
30
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
How to Read the Table …
31
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Conclusion
Intrinsic carrier concentration is so small that semiconductor
must be doped to make it useful.
A doping atom behaves like a H-atom, except that the
dielectric constant and effective masses are given by by those
of the host atom.
32
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Law of mass-action & intrinsic concentration •Statistics of donors and acceptor levels• Intrinsic carrier concentration•Temperature dependence of carrier concentration•Multiple doping, co-doping, and heavy-doping•Conclusions
33
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Looking ahead: Carrier-Density w/Doping
0−+ − + − = ∫ D ANp n dVN
0+ −− + − =ADp n NN
1 10
42− − −
− −−
++ =
+− −
F D
F
A B
C
F
B F B
B
V D( E E )
A( E E ) / k T ( E E )( E
/ k TE ) / k TC / k TVN
NN
eee
Ne
A bulk material must be charge neutral over all …
Further if the material is spatially homogenous
Let us see how the formula come about …
34
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Characteristics of Donor Atoms
(D)
r0
ET
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Localized vs. Delocalized States
2N states/per-band (with spin)
2N-2 states/per-band (with spin)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Statistics of Donor Levels
0 0 0 0 1
1 0 1 1
0 1 1 1
i F
B
i F
B
i i i
( E E )
k T
( E E )
k T
u / d E N P
/ / Z
/ e Z
/ e Z
−−
−−
1/1 x x x
− − − −
− −= ≡∑
i i F B F B
i i F
ii
B
( E N E ) / k T ( E ) / k T
i ( E N E ) /
N
k T
i
Ee eP
e Z
Coulomb interaction forbids this configuration
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Statistics of Donor Levels
0 0 0 0 1
0 1 1 1
1 0 1 1
i F
B
i F
B
i i i
( E E )
k T
( E E )
k T
u / d E N P
/ / Z
/ e Z
/ e Z
−−
−−
0000
00 01 10 2
1 1
1 2 1− − −= = =+ + + +i F B F i B( E E ) / k T ( E E ) / k T
P / Zf
P P P / Z e / Z e
00
1 11 1
1 11
22 −
−− = − =
+ +F i B
i F B
( E E )/ k T( E E ) / k T
fe e
Prob. that the donoris empty (charged)
Prob. that the donoris filled with at leastone electron (neutral)
Note the extra factor ….
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Coulomb Exclusion for Band Electrons?
0 0 0 1
1 1 1i F
B
( E E
i
)
i
T
i
k
state E N P
/ Z
e Z−
−
i i F B i i F B
i i F B
( E N E ) / k T ( E N E ) / k T
i ( E N E ) / k T
i
e eP
e Z
− − − −
− −= ≡∑
Ei
E=0
E=2E=4
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Localized vs. Band Electrons
E1 ← 2π/Lx
E2 ← 4π/Lx
E3 ← 6π/Lx
E4 ← 8π/Lx
E5 ← 10π/Lx
E6 ← 12π/Lx
E1’ ← 2p/(Lx/2)
E2’ ← 4π/(Lx/2)
E3’ ← 6π/(Lx/2)
Lx
Lx/2 Lx/2
Two electrons (even with opposite spin) can not be at the same position and same energy becauseof electrostatic repulsion
Band electrons (with opposite spin) need not be at the same position, so they can share occupy same energy level.
When we divide space by a factor of 2, the number of states (e.g. 6 here) does not change.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Acceptor Atoms
State [1] …. Hole present … N-1 charges
State [0] … Hole filled …. N charges
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Statistics of Acceptor Levels in Si and Ge
1. Each atom contributes 2 states (up & down spin) to a band, therefore a band has 2N states.
2. Every time a host atom is replaced by a impurity atom, 2 states are disappear per a band and appear as localized states (sort of).
3. Therefore an acceptor atom close to hh and lh bands removes four states from those bands.
4. Because of Coulomb interaction only 1 hole can seat in these 4 states: the states are 0000, 0001, 0010, 0100, 1000.
5. One now uses P to compute the occupation
from lhfrom hh
EC
EV
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Number and Energy Considerations …
4N-4 StatesIn HH/LH bands
0000 1000 0100 0010 0001
1) [0000] is the charged state as it has N electrons, but N-1 protons.
2) Single hole configuration [0001] is uncharged, as we have N-1 electrons, and N-1 protons … same is true for [0010], [0100], [1000] states.
3) Going from [0000] to [0001] states, the number of electrons goes down by 1 (Ni=-1).
4) Going from [0000] to [0001] states energy goes down by –EA, because one electron is no longer occupying the high energy level at EA.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Statistics of Acceptor Levels
0000 1000 0100 0010 00010 0
0000
1F B
i i F B
( xE ) / k T
( E N E ) / k T
i
eP
e Z
− −
− −= ≡∑
1
0001 0010 0100 1000
A F B A F B
i i F B
( E ( )E ) / k T ( E E ) / k T
( E N E ) / k T
i
e eP P P P
e Z
− − − − −
− −= = = = ≡∑
Steps 3 & 4
00000000
0000 1000 0100 0010 0001
1
1 4 A F B( E E ) / k T
Pf
P P P P P e −= =+ + + + +
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Filled and empty Donor/Acceptor Levels
00
1
1 2 −= =+ F i BD D ( E E ) / k T
N f Ne
[ ]0000−≡ =filled
A A AN N N f
2N-2 states
4N-4 StatesIn HH/LH bands
(Two holes can not seat together)
0000 1000 0100 0010 0001
00 10 01 +≡emptyD DN N
1
1 4 −=+ A F BA ( E E ) / k T
Ne
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Distributions are physical ….
1 −=+ F D B
DD ( E / T
DE ) kg
Nf
e
1 1ε − −
= =+ +F D B F
'B D B
D DD ( E E ) / kk T ET ( E ) / k T
N Nf
e e e
Degeneracy factor …
Effective donor level
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Summary …
0ADNp n dVN−+ − + + = ∫
0AD Nn Np + −− + + =
10
12 4F V B C F B
F D A F BB
( E E ) / k T ( E E ) / k T A( E E ) /V A
D( E E ) k/ k TT
Ne e
eN N
N
e− − −
−−
−+ +− + − =
A bulk material must be charge neutral over all …
Further if the material is spatially homogenous
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Law of mass-action & intrinsic concentration •Statistics of donors and acceptor levels• Intrinsic carrier concentration•Temperature dependence of carrier concentration•Multiple doping, co-doping, and heavy-doping•Conclusions
48
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Carrier-density with Uniform Doping
0D Ap n N N dV+ − − + + = ∫
0D Ap n N N+ −− + + =
01 2 1 4
− − − −− −− + − =
+ +V B C B
D B A
F F
F BF
( E ) / k T ( E ) / k TV A ( E ) / k T ( E
E EE E )
D/ k T
AN e N ee e
N N
A bulk material must be charge neutral over all …
Further if the doping is spatially homogenous
Once you know EF, you can calculate n, p, ND+, NA
-.
( ) ( )1 2 1 2
2 20
1 2 1 4β ββ βπ π − − − − − + − = + +F FD AF F E EV V A C ( E )
D)
A( E
NN F E E
NN F
e eE E
(approx.)
FD integral vs. FD function ?
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Intrinsic Concentration
0D Ap n N N+ −− + + =
01 2 1 4
− − − −− −− + − =
+ +F V B C F B
F D B A F B
( E E ) / k T ( E E ) / k T( E E ) / k T ( E E )V C
Ak T
D/
N
eNN e
Ne
e
( ) ( )0 β β− − + −− = ⇒ =c F v FE E EV
ECn p e eN N
1
2 2β≡ = + VG
F iC
EE E l
N
Nn
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Law of mass-action & intrinsic concentration •Statistics of donors and acceptor levels• Intrinsic carrier concentration•Temperature dependence of carrier concentration•Multiple doping, co-doping, and heavy-doping•Conclusions
51
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Carrier Density with Donors
0D Ap n N N+ −− + + =
01 2 1 4
− − − −− −− + − =
+ +F V B C F B
F D B A F B
( E E ) / k T ( E E ) / k T A( E E ) / k T ( E E ) / k T
DCVN
Ne e
e eN
N
In spatially homogenous field-free region …
Assume N-type doping …
n(will plot in next slide)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Temperature-dependent Concentration
D
n
N
Temperature
1
Freezeout Extrinsic
Intrinsic
i
D
n
N
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Physical Interpretation
D
n
N
Temperature
1
Freezeout Extrinsic
Intrinsic
i
D
n
N
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Electron Concentration with Donors
1 2 β+
−=+ DFD E
D( E )
Ne
N
β β β− −= ⇒ =FC FC( E )E E
C
EC
n
Nn e e eN
1 2 β −
=
+
c D
D
( E E )
C
eN
N
n 1ξ
≡+
DNn
N
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Electron concentration with Donors
0+− + =Dp n N
01 2
F V B C F B
F D B
( E E ) / k T ( E E ) / k T DV C ( E E ) / k T
NN e N e
e− − − −
−− + =+
2
01
ξ
− + =+
i Dnn
N
N
nn
No approximation so far ….
2× = ip n n
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
High Donor density/Freeze out T
2
01
ξ
− + =+
i Dnn
N
N
nn
D iN n≫
2
01
0
ξ
ξ ξ
⇒ − + ≈+
⇒ + − =
D
D
N
N
N N
n
N
n
n n1 2
41 1
2ξ
ξ
= + −
DN
nN
N
D
n
N
Temperature
1
Freezeout Extrinsic
Intrinsic
i
D
n
N
( )2ξ
β− −≡ C DE ECNN e
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Extrinsic T
( )2
C DE E / kTCD
NN e Nξ
− −≡ ≫
1 2
41 1
2ξ
ξ
= + −
DN
nN
N
Electron concentration equals donor densityhole concentration by nxp=ni2
D
n
N
Temperature
1
Freezeout Extrinsic
Intrinsic
i
D
n
N
41
21 1
2ξ
ξ
≈ + −
DNN
N
≈ DN
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Intrinsic T
2
01
ξ
− + =+
i Dnn
N
Nn
n
1
2
22
2 4
= + +
i
D D nnN N
2 for
1+
−= ≈ <+ F D B( E E ) /
DDk TD F D
NN E
eN E
D
n
N
Temperature
1
Freezeout Extrinsic
Intrinsic
i
D
n
N
0
2
0− + ≈iD
nn N
n
2 2 0⇒ − + − =i Dn n N n
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Extrinsic/Intrinsic T
For i Dn N≫
1 222
2 4
= + + ≈
i i
D Dn nN N
n
2 1 2
2
2 4
= + + ≈
i
D DDn n
N NN
For D iN n≫
D
n
N
Temperature
1
Freezeout Extrinsic
Intrinsic
i
D
n
N
What will happen if you use silicon circuits at very high temperatures ?
Bandgap determines the intrinsic carrier density.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Determination of Fermi-level
( ) 1β
β− −
= ⇒ = +
FcEC C
EF
C
N e EN
nlnn E
0Dp n N+− + =
01 2
− − − −−− + =
+F FV B C B
D BF
( E ) / k T ( E ) / k T DV
E E( kEC E ) / T
NN e N e
e
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Law of mass-action & intrinsic concentration •Statistics of donors and acceptor levels• Intrinsic carrier concentration•Temperature dependence of carrier concentration•Multiple doping, co-doping, and heavy-doping•Conclusions
62
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Multiple Donor Levels
210
1 2 1 2 1 4− − −− + + − =+ + +DF B F B A F BD
D D A( E ) / k T ( E ) / k T ( E EE kE ) / T
N N Np n
e e e
Multiple levels of same donor …
1 2
1 2 01 2 1 2 1 4− − −− + + − =
+ + +DF B F B AD F B
D A( E ) / k T ( E ) / k T ( E
DE E ) / k TE
N Np n
e e
N
e
Codoping…
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Heavy Doping Effects: Bandtail States
E
k
k ?
E
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Heavy Doping Effects: Hopping Conduction
E
k
Bandgap narrowing
β−× =*
GC V
Ep n N N e
Band transport vs.
hopping-transport
e.g. Base of HBTsK?
E
e.g. a-silicon, OLED
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Arrangement of Atoms
Poly-crystallineThin Film Transistors
Crystalline
AmorphousOxides
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Poly-crystalline material
Isotropic bandgap and increase in scattering
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Band-structure and Periodicity
Periodicity is sufficient, but not necessary for bandgap. Many amorphous material show full isotropic bandgap
PR
B, 4
, 250
8, 1
971
Eda
gaw
a, P
RL,
100
,013
901,
200
8
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Conclusions
1. Charge neutrality condition and law of mass-action allows calculation of Fermi-level and all carrier concentration.
2. For semiconductors with field, charge neutrality will not hold and we will need to use Poisson equation.
3. Heaving doping effects play an important role in carrier transport.